The course looks fantastic. I have plenty of experience in R, Rstudio, git, and GitHub, which are the main tools of my daily work. With this course I hope to not only get hands on with the selection and application of various statistical techniques, but also gain a better outlook on the best practices to visualize and share data.
date()
## [1] "Fri Dec 03 11:07:27 2021"
Github repository: https://github.com/SchahzadSaqib/IODS-project
date()
## [1] "Fri Dec 03 11:07:27 2021"
library(tidyverse)
#' read in data for analysis
lrn14 <- read.csv(here::here("data",
"lrn14_summ.csv")) %>%
dplyr::select(gender, everything()) #' reorder columns
#' take a look at the data
dplyr::glimpse(lrn14)
## Rows: 166
## Columns: 7
## $ gender <chr> "F", "F", "F", "F", "F", "F", "F", "F", "F", "F", "M", "F", "~
## $ Age <int> 17, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 2~
## $ Attitude <dbl> 1.7, 3.2, 3.7, 3.8, 1.9, 2.0, 3.2, 3.6, 3.7, 4.0, 1.9, 1.9, 2~
## $ Points <int> 26, 17, 18, 28, 23, 20, 24, 30, 18, 27, 12, 12, 21, 20, 19, 2~
## $ deep <dbl> 3.916667, 4.500000, 3.166667, 3.166667, 3.250000, 4.083333, 4~
## $ stra <dbl> 4.625, 3.375, 2.625, 4.000, 3.875, 3.375, 3.625, 2.625, 2.625~
## $ surf <dbl> 3.416667, 3.166667, 3.416667, 2.250000, 3.000000, 2.833333, 2~
The data comes from the international survey of Approaches to Learning, supported by the Teachers’ Academy funding for KV in 2013-2015. The data consists of 166 individuals and 7 variables. There are two descriptive variables: age and gender, and 5 variables that represent the cumulative results of the questionnaires in the survey. These are “points” (total exam points for each individual), “attitude” (global attitude towards statistics, cumulative sum of 10 questions ranking from 1-5), “deep” (a set of questions targeting the deep approach to learning), “stra” (strategic approach to learning), and “surf” (surface approach to learning).
Further details and explanations regarding this dataset can be found here
library(GGally)
#' plot grahic summary of variables
GGally::ggpairs(lrn14,
mapping = aes(col = gender,
fill = gender,
alpha = 0.2),
lower = list(combo = GGally::wrap("facethist",
bins = 20))) +
#' assign fill and colour
viridis::scale_colour_viridis(option = "G",
end = 0.8,
discrete = T) +
viridis::scale_fill_viridis(option = "G",
end = 0.8,
discrete = T)
From the descriptive variables we can observe that there was a higher proportion of females in the participants and that the average age of the participants was ~20-25. There are two pairs of variables that correlate: 1) Points and attitude: 0.422, which shows that higher exam points are correlated to better attitude towards statistics and 2) deep and surf: -0.324, which shows that deep learning approaches are negatively correlated with surface approaches.
Using exam points (points) as our dependent variable, we will fit a linear model to determine which independent variables are most crucial in explain the distribution of exam points in the dataset. Attitude is the first choice since it was seen to be highly correlated with exam points. We also add the variables surf and stra to the model since they are also seen to be marginally correlated.
#' linear model
lm_mdl <- stats::lm(formula = Points ~ Attitude + surf + stra,
data = lrn14)
#' model summary
summary(lm_mdl)
##
## Call:
## stats::lm(formula = Points ~ Attitude + surf + stra, data = lrn14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.1550 -3.4346 0.5156 3.6401 10.8952
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.0171 3.6837 2.991 0.00322 **
## Attitude 3.3952 0.5741 5.913 1.93e-08 ***
## surf -0.5861 0.8014 -0.731 0.46563
## stra 0.8531 0.5416 1.575 0.11716
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.296 on 162 degrees of freedom
## Multiple R-squared: 0.2074, Adjusted R-squared: 0.1927
## F-statistic: 14.13 on 3 and 162 DF, p-value: 3.156e-08
As expected, attitude is statistically significant (p-value 1.93e-08). The variables surf and stra however so seem to be significant. This model explains ~19% (R2 0.1927) of the exam point distribution. Next, we will remove the non-significant variables from the model.
#' linear model
lm_mdl <- stats::lm(formula = Points ~ Attitude,
data = lrn14)
#' summary
summary(lm_mdl)
##
## Call:
## stats::lm(formula = Points ~ Attitude, data = lrn14)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.6372 1.8303 6.358 1.95e-09 ***
## Attitude 3.5255 0.5674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09
Our new model still explains ~19% of the distribution (R2 = 0.1856) and attitude is statistically significant.
Next, we validate our model by plotting some diagnostics plots
#' diagnostic plots
par(mfrow = c(3, 1))
plot(lm_mdl, which = c(1,2,5))
Residuals vs Fitted plot: These plots illustrated the patterns of the residuals. While using a linear model, it is crucial to know that the residuals to not exhibit any non-linear patterns which would need alternative model to capture. The residuals look fairly linear in our plot, which would indicated that the linear model was the correct choice.
Normal Q-Q plot: The plot illustrates whether the residuals are normally distributed. They should roughly follow the vertical line, with some room for deviations in the tail end. These residuals seem normally distributed with a few potential outliers.
Residuals vs Leverage plot: The plot indicates whether there are standout samples that are highly influential towards the overall distribution and regression analysis reuslts. Such cases/samples would cross dashed lines in the upper and lower right corners of the plot, which represent the Cooks distance. No such instances are observed in our plots, which means there are no highly influential samples present in our data.
This data set comes from a Portuguese study titled “Using data mining to predict secondary school student performance”. The data is collected from two Portuguese secondary schools and consists of descriptive attributes of students personal lives as well their performance in school, measured by their average grade performance. Further details about the study and the data set can be seen here and here.
Lets take a look at the data itself
#' load main library
library(tidyverse)
#' read in data
stu_aggr <- read.csv(here::here("data",
"stu_aggr.csv"))
#' check the structure and dimensions
dplyr::glimpse(stu_aggr)
## Rows: 370
## Columns: 47
## $ school <chr> "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP~
## $ sex <chr> "F", "F", "F", "F", "F", "M", "M", "F", "M", "M", "F", "F~
## $ age <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15, 15, 1~
## $ address <chr> "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U~
## $ famsize <chr> "GT3", "GT3", "LE3", "GT3", "GT3", "LE3", "LE3", "GT3", "~
## $ Pstatus <chr> "A", "T", "T", "T", "T", "T", "T", "A", "A", "T", "T", "T~
## $ Medu <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, 3, 3, ~
## $ Fedu <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, 3, 2, ~
## $ Mjob <chr> "at_home", "at_home", "at_home", "health", "other", "serv~
## $ Fjob <chr> "teacher", "other", "other", "services", "other", "other"~
## $ reason <chr> "course", "course", "other", "home", "home", "reputation"~
## $ guardian <chr> "mother", "father", "mother", "mother", "father", "mother~
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, ~
## $ studytime <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, 2, 1, ~
## $ failures.mat <int> 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, ~
## $ schoolsup <chr> "yes", "no", "yes", "no", "no", "no", "no", "yes", "no", ~
## $ famsup <chr> "no", "yes", "no", "yes", "yes", "yes", "no", "yes", "yes~
## $ paid.mat <chr> "no", "no", "yes", "yes", "yes", "yes", "no", "no", "yes"~
## $ activities <chr> "no", "no", "no", "yes", "no", "yes", "no", "no", "no", "~
## $ nursery <chr> "yes", "no", "yes", "yes", "yes", "yes", "yes", "yes", "y~
## $ higher <chr> "yes", "yes", "yes", "yes", "yes", "yes", "yes", "yes", "~
## $ internet <chr> "no", "yes", "yes", "yes", "no", "yes", "yes", "no", "yes~
## $ romantic <chr> "no", "no", "no", "yes", "no", "no", "no", "no", "no", "n~
## $ famrel <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, 5, 5, ~
## $ freetime <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, 3, 5, ~
## $ goout <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, 2, 5, ~
## $ Dalc <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, ~
## $ Walc <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 4, ~
## $ health <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, 4, 5, ~
## $ absences.mat <int> 6, 4, 10, 2, 4, 10, 0, 6, 0, 0, 0, 4, 2, 2, 0, 4, 6, 4, 1~
## $ G1.mat <int> 5, 5, 7, 15, 6, 15, 12, 6, 16, 14, 10, 10, 14, 10, 14, 14~
## $ G2.mat <int> 6, 5, 8, 14, 10, 15, 12, 5, 18, 15, 8, 12, 14, 10, 16, 14~
## $ G3.mat <int> 6, 6, 10, 15, 10, 15, 11, 6, 19, 15, 9, 12, 14, 11, 16, 1~
## $ failures.por <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, ~
## $ paid.por <chr> "no", "no", "no", "no", "no", "no", "no", "no", "no", "no~
## $ absences.por <int> 4, 2, 6, 0, 0, 6, 0, 2, 0, 0, 2, 0, 0, 0, 0, 6, 10, 2, 2,~
## $ G1.por <int> 0, 9, 12, 14, 11, 12, 13, 10, 15, 12, 14, 10, 12, 12, 14,~
## $ G2.por <int> 11, 11, 13, 14, 13, 12, 12, 13, 16, 12, 14, 12, 13, 12, 1~
## $ G3.por <int> 11, 11, 12, 14, 13, 13, 13, 13, 17, 13, 14, 13, 12, 13, 1~
## $ G1 <int> 2, 7, 10, 14, 8, 14, 12, 8, 16, 13, 12, 10, 13, 11, 14, 1~
## $ G2 <int> 8, 8, 10, 14, 12, 14, 12, 9, 17, 14, 11, 12, 14, 11, 15, ~
## $ G3 <int> 8, 8, 11, 14, 12, 14, 12, 10, 18, 14, 12, 12, 13, 12, 16,~
## $ absences <int> 5, 3, 8, 1, 2, 8, 0, 4, 0, 0, 1, 2, 1, 1, 0, 5, 8, 3, 9, ~
## $ failures <int> 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, ~
## $ paid <chr> "no", "no", "yes", "yes", "yes", "yes", "no", "no", "yes"~
## $ alc_use <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1.5, 1.~
## $ high_use <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, FA~
As we can see, there are 370 student, and 47 characteristics defined for each one. The most important are G3, which represents their final grade and high_use, which is measure of their alcohol intake during the week. Our research question is whether an increase in alcohol intake is associated to the students overall performance and/or is significantly associated to certain variables.
Lets first take a look at the distribution of the variables:
#' removing duplicated and aggregated columns
stu_aggr_clean <- stu_aggr %>%
dplyr::select(!ends_with(c(".mat", ".por")),
-Dalc,
-Walc)
#' plot distributions
plot_vars <- stu_aggr_clean %>%
gather() %>%
ggplot2::ggplot(., aes(value, fill = key)) +
ggplot2::facet_wrap(key ~., scales = "free") +
ggplot2::geom_bar(show.legend = F) +
ggplot2::theme_minimal() +
viridis::scale_fill_viridis(option = "F",
discrete = T)
plot_vars
Looking at these distributions, we can select 4 variables that could potentially play a role in increased alcohol intake and see whether they come out to be statistically significant. Lets pick the following:
goout: The frequency of going out with friends (1 - very low to 5 - very high).
sex: students sex (M - male, F - female)
famrel: quality of family relationships (1 - very bad to 5 - very good)
paid: extra paid classes within the course subject (Math or Portuguese)
vars_sel <- c("goout",
"famrel",
"sex",
"paid")
to.plot <- stu_aggr_clean %>%
dplyr::select(vars_sel[1:2], high_use) %>%
tidyr::pivot_longer(cols = -high_use) %>%
dplyr::arrange(name) %>%
dplyr::mutate(name = factor(name, levels = c(unique(name))))
ggplot2::ggplot(to.plot, aes(x = high_use,
y = value,
fill = high_use)) +
ggplot2::facet_wrap(name ~., scales = "free") +
ggplot2::geom_jitter(aes(colour = high_use),
show.legend = F) +
ggplot2::geom_boxplot(show.legend = F) +
viridis::scale_fill_viridis(option = "G",
begin = 0.4,
end = 0.95,
discrete = T) +
viridis::scale_colour_viridis(option = "G",
begin = 0.4,
end = 0.95,
discrete = T) +
ggplot2::theme_minimal()
to.plot <- stu_aggr_clean %>%
dplyr::select(vars_sel[3:4], high_use) %>%
tidyr::pivot_longer(cols = -high_use) %>%
dplyr::arrange(name) %>%
dplyr::mutate(name = factor(name, levels = c(unique(name))))
ggplot2::ggplot(to.plot, aes(x = high_use)) +
ggplot2::facet_wrap(name ~ ., scales = "free") +
ggplot2::geom_bar(aes(fill = value)) +
viridis::scale_fill_viridis(option = "G",
discrete = T,
begin = 0.4,
end = 0.9) +
ggplot2::theme_minimal()
for (sub in 1:length(vars_sel)) {
to.summ <- stu_aggr_clean %>%
dplyr::group_by(high_use, !!!syms(vars_sel[sub])) %>%
dplyr::summarise(n = n(), mean_G3 = mean(G3)) %>%
dplyr::arrange(!!!vars_sel[sub])
print(to.summ)
}
## `summarise()` has grouped output by 'high_use'. You can override using the `.groups` argument.
## # A tibble: 10 x 4
## # Groups: high_use [2]
## high_use goout n mean_G3
## <lgl> <int> <int> <dbl>
## 1 FALSE 1 19 10.7
## 2 FALSE 2 82 12.2
## 3 FALSE 3 97 12.2
## 4 FALSE 4 40 11.5
## 5 FALSE 5 21 10.1
## 6 TRUE 1 3 11.3
## 7 TRUE 2 15 11.9
## 8 TRUE 3 23 10.9
## 9 TRUE 4 38 11.1
## 10 TRUE 5 32 10.0
## `summarise()` has grouped output by 'high_use'. You can override using the `.groups` argument.
## # A tibble: 10 x 4
## # Groups: high_use [2]
## high_use famrel n mean_G3
## <lgl> <int> <int> <dbl>
## 1 FALSE 1 6 11.8
## 2 FALSE 2 9 11.2
## 3 FALSE 3 39 11.4
## 4 FALSE 4 128 11.7
## 5 FALSE 5 77 12.2
## 6 TRUE 1 2 12
## 7 TRUE 2 9 10.7
## 8 TRUE 3 25 10.8
## 9 TRUE 4 52 11.1
## 10 TRUE 5 23 10.2
## `summarise()` has grouped output by 'high_use'. You can override using the `.groups` argument.
## # A tibble: 4 x 4
## # Groups: high_use [2]
## high_use sex n mean_G3
## <lgl> <chr> <int> <dbl>
## 1 FALSE F 154 11.4
## 2 FALSE M 105 12.3
## 3 TRUE F 41 11.8
## 4 TRUE M 70 10.3
## `summarise()` has grouped output by 'high_use'. You can override using the `.groups` argument.
## # A tibble: 4 x 4
## # Groups: high_use [2]
## high_use paid n mean_G3
## <lgl> <chr> <int> <dbl>
## 1 FALSE no 140 11.4
## 2 FALSE yes 119 12.3
## 3 TRUE no 56 10.8
## 4 TRUE yes 55 11.0
As we can see from the plots and tables, low family relations and high frequency of going out with friends are associated with increased alcohol consumption. Males were also more likely to consume more alcohol then females. Extra paid courses does not seem to show any trends, but may become clearer within the model.
Lets use all 4 variables in a logistic model for our response variable - high_use
#' fit model
glm.1 <- stats::glm(high_use ~ sex + goout + famrel + paid,
data = stu_aggr_clean,
family = "binomial")
#' summarise
summary(glm.1)
##
## Call:
## stats::glm(formula = high_use ~ sex + goout + famrel + paid,
## family = "binomial", data = stu_aggr_clean)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.6862 -0.7894 -0.5213 0.8510 2.5063
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -2.4751 0.6680 -3.705 0.000211 ***
## sexM 1.0333 0.2596 3.981 6.88e-05 ***
## goout 0.8022 0.1231 6.517 7.20e-11 ***
## famrel -0.4350 0.1404 -3.098 0.001950 **
## paidyes 0.3161 0.2562 1.234 0.217272
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.04 on 369 degrees of freedom
## Residual deviance: 377.58 on 365 degrees of freedom
## AIC: 387.58
##
## Number of Fisher Scoring iterations: 4
#' extract coefficients
coef(glm.1)
## (Intercept) sexM goout famrel paidyes
## -2.4751385 1.0333396 0.8021770 -0.4349508 0.3161099
#' extract Odds ratios
OR <- coef(glm.1) %>%
exp
#' extract confidence intervals
CI <- confint(glm.1) %>%
exp
## Waiting for profiling to be done...
#' create table
cbind(OR, CI)
## OR 2.5 % 97.5 %
## (Intercept) 0.08415133 0.02186678 0.3024711
## sexM 2.81043581 1.70049874 4.7152163
## goout 2.23039118 1.76438158 2.8618618
## famrel 0.64729650 0.48962135 0.8506400
## paidyes 1.37178094 0.83155189 2.2751637
As expected, sex (male), going out with friends, and family relations are important factors in explaining increased alcohol consumption. There is a positive relation with high_use and going out and a negative relation between high_use and better family relations.
The odds ratios is the probability of success over probability of failure. Once again, these show the higher proportion of alcohol consumption among males, going out, and decreasing family relations.
Next, we predict the results (high_use) using our model and observe its accuracy it assigning the correct outcomes.
#' predict and subset
stu_aggr_wprobs <- stu_aggr_clean %>%
dplyr::mutate(probs = predict(glm.1, type = "response"),
preds = probs > 0.5) %>%
dplyr::select(high_use, !!!vars_sel, probs, preds)
tail(stu_aggr_wprobs, 10)
## high_use goout famrel sex paid probs preds
## 361 FALSE 3 4 M no 0.31538311 FALSE
## 362 FALSE 2 4 M no 0.17118555 FALSE
## 363 TRUE 3 5 M yes 0.29030314 FALSE
## 364 FALSE 3 5 F yes 0.12705507 FALSE
## 365 FALSE 3 4 F yes 0.18357661 FALSE
## 366 FALSE 2 5 F no 0.04541047 FALSE
## 367 FALSE 4 4 F yes 0.33400549 FALSE
## 368 FALSE 1 1 F no 0.10833016 FALSE
## 369 TRUE 5 2 M no 0.84542817 TRUE
## 370 TRUE 1 4 M no 0.08475514 FALSE
#' confusion matrix
table(high_use = stu_aggr_wprobs$high_use,
preds = stu_aggr_wprobs$preds)
## preds
## high_use FALSE TRUE
## FALSE 235 24
## TRUE 62 49
Lets now determine the predictive power of our model against randomly guessing the outcome. To do this we calculate the mean incorrectly classified outcomes with different probabilities. This serves as a penalty and the lower this value the better.
#' defining the loss function
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
#' probabiltiy of high use is 1 for each individual
loss_func(class = stu_aggr_wprobs$high_use,
prob = 1)
## [1] 0.7
#' probabiltiy of high use is 0 for each individual
loss_func(class = stu_aggr_wprobs$high_use,
prob = 0)
## [1] 0.3
#' probabiltiy of high use are taken from the model for each
#' individual
loss_func(class = stu_aggr_wprobs$high_use,
prob = stu_aggr_wprobs$probs)
## [1] 0.2324324
So, our model has the lowest penalty compared to randomly guessing.
Next, lets see how our model performs on randomized subsets of our data, once again measuring the mean incorrectly classified outcomes as the measure for performance
#' cross validation
library(boot)
#' cross-validation
stu_aggr_cv <- boot::cv.glm(data = stu_aggr_wprobs,
cost = loss_func,
glmfit = glm.1,
K = 10)
stu_aggr_cv$delta[1]
## [1] 0.2378378
The prediction errors are slightly higher but it stills performs better than guessing.
This weeks exercise is all about visually exploring statistical data. The exercise will be based on the Boston data provided with the MASS R package.
First, load the MASS library and read in the data:
library(tidyverse)
library(MASS)
##
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
##
## select
BSTN <- MASS::Boston
glimpse(BSTN)
## Rows: 506
## Columns: 14
## $ crim <dbl> 0.00632, 0.02731, 0.02729, 0.03237, 0.06905, 0.02985, 0.08829,~
## $ zn <dbl> 18.0, 0.0, 0.0, 0.0, 0.0, 0.0, 12.5, 12.5, 12.5, 12.5, 12.5, 1~
## $ indus <dbl> 2.31, 7.07, 7.07, 2.18, 2.18, 2.18, 7.87, 7.87, 7.87, 7.87, 7.~
## $ chas <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,~
## $ nox <dbl> 0.538, 0.469, 0.469, 0.458, 0.458, 0.458, 0.524, 0.524, 0.524,~
## $ rm <dbl> 6.575, 6.421, 7.185, 6.998, 7.147, 6.430, 6.012, 6.172, 5.631,~
## $ age <dbl> 65.2, 78.9, 61.1, 45.8, 54.2, 58.7, 66.6, 96.1, 100.0, 85.9, 9~
## $ dis <dbl> 4.0900, 4.9671, 4.9671, 6.0622, 6.0622, 6.0622, 5.5605, 5.9505~
## $ rad <int> 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4,~
## $ tax <dbl> 296, 242, 242, 222, 222, 222, 311, 311, 311, 311, 311, 311, 31~
## $ ptratio <dbl> 15.3, 17.8, 17.8, 18.7, 18.7, 18.7, 15.2, 15.2, 15.2, 15.2, 15~
## $ black <dbl> 396.90, 396.90, 392.83, 394.63, 396.90, 394.12, 395.60, 396.90~
## $ lstat <dbl> 4.98, 9.14, 4.03, 2.94, 5.33, 5.21, 12.43, 19.15, 29.93, 17.10~
## $ medv <dbl> 24.0, 21.6, 34.7, 33.4, 36.2, 28.7, 22.9, 27.1, 16.5, 18.9, 15~
The data consists of 506 entries for 14 variables that describe the housing value of different suburbs of Boston, New York, the air quality, and willingness of occupants to pay for clean air. A detailed description of each variable can be seen here
summary(BSTN)
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
It seems like there is a lot of variation between different towns/suburbs, with the widest ranges being present in crime rates per capita (min: 0.0062, max: 88.98), age (min: 2.9, max: 100), and proportion of residential land zoned for lots over 25,000 sq.ft. (min: 0, max: 100).
Now lets look at the correlations between these variables:
#' create correlation matrix
BSTN_cor <- cor(BSTN) %>%
round(digits = 2)
#' visualize
corrplot::corrplot(BSTN_cor,
type = "upper",
tl.pos = "d")
As can be observed from the plot, there is high correlation between most of the variables, with the most striking being the positive correlation of nox vs age, negative correlation of nox vs indus, and positive correlation of rad vs tax.
To account for the massive ranges of certain variable and to keep the data comparable, lets scale the variables to center the means around 0.
BSTN_scale <- as.data.frame(scale(BSTN))
summary(BSTN_scale)
## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865
Next, let restructure the crime rate variable to a categorical variable:
#' generate quantiles
crime_qtls <- quantile(BSTN_scale$crim)
#' create categroical variable
BSTN_scale_mod <- BSTN_scale %>%
dplyr::mutate(crime = cut(crim,
breaks = crime_qtls,
include.lowest = T,
labels = c("low",
"mid_low",
"mid_high",
"high"))) %>%
dplyr::select(-crim)
With crime as the variable of interest, we divide the data set into training and test sets
#' randomly extract indexes for 80% of the data
sub <- sample(nrow(BSTN_scale_mod),
size = nrow(BSTN_scale_mod) * 0.8)
#' subset the training set (80%)
train <- BSTN_scale_mod[sub,]
#' subset the test set (20%)
test <- BSTN_scale_mod[-sub,]
#' save correct classes
crt <- test$crime
#' remove crime from test data set
test <- test %>%
dplyr::select(-crime)
Next, we perform Linear discriminant analysis (LDA) on the training set, which is a dimensionality reduction technique to identify separation within the data based on features/variables.
#' LDA
BSTN_lda <- MASS::lda(crime ~ ., data = train)
BSTN_lda
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low mid_low mid_high high
## 0.259901 0.240099 0.259901 0.240099
##
## Group means:
## zn indus chas nox rm age
## low 1.0348982 -0.9197802 -0.12234430 -0.8997404 0.4684579 -0.9298066
## mid_low -0.1146069 -0.2453174 0.01179157 -0.5350933 -0.1108013 -0.3052129
## mid_high -0.3794348 0.1103209 0.17762524 0.3243173 0.0918388 0.3735702
## high -0.4872402 1.0172187 0.01179157 1.0718359 -0.4373125 0.8398408
## dis rad tax ptratio black lstat
## low 0.9238427 -0.6909262 -0.7214972 -0.47963828 0.3768695 -0.793148242
## mid_low 0.3029008 -0.5343243 -0.4681280 -0.05327035 0.3151947 -0.137244799
## mid_high -0.3320558 -0.3857621 -0.3098880 -0.25484432 0.1048269 -0.005563586
## high -0.8569352 1.6371072 1.5133254 0.77958792 -0.7622250 0.900329437
## medv
## low 0.5203038770
## mid_low 0.0009166787
## mid_high 0.1684332832
## high -0.6605420910
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.08822181 0.84339651 -0.76731376
## indus -0.02303195 -0.26733247 0.44886061
## chas -0.05790445 -0.05886530 0.07567103
## nox 0.43386681 -0.54050924 -1.55904267
## rm -0.09850735 -0.06472423 -0.15348683
## age 0.25654789 -0.28719825 -0.17421549
## dis -0.08756626 -0.22615601 0.02444811
## rad 3.01699904 0.80988189 0.17764119
## tax -0.06224867 0.12039532 0.32972045
## ptratio 0.12295211 0.04893984 -0.27694434
## black -0.12552205 -0.01878472 0.11383890
## lstat 0.23775802 -0.28463889 0.32446697
## medv 0.20054721 -0.39729101 -0.25681264
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9432 0.0440 0.0128
#' define functions for lda arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
#' target classes
tr_crt <- as.numeric(train$crime)
#' plot LDA
plot(BSTN_lda,
col = tr_crt,
dimen = 2,
pch = tr_crt)
lda.arrows(BSTN_lda, myscale = 2)
As we can see from the plot, crime rates separate very well between high and mid-low/low, while there is still some separation between the mid_high and low/mid_low categories. Furthermore, the variables rad, nox, and zn seem to be crucial for the separation.
Now lets predict the values on the test data set
#' predict the classes
BSTN_pred <- predict(BSTN_lda,
newdata = test)
#' compare against the correct classes
table(correct = crt,
predicted = BSTN_pred$class)
## predicted
## correct low mid_low mid_high high
## low 9 11 2 0
## mid_low 5 20 4 0
## mid_high 0 1 20 0
## high 0 0 0 30
There were some classification errors in the low class, but they were fairly good for the rest.
Next up, we find the distance between the variables and perform k-means clustering
#' distances
BSTN_dist <- stats::dist(BSTN_scale)
#' summary
summary(BSTN_dist)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970
#' manhattan distance matrix
BSTN_dist_man <- dist(BSTN_scale,
method = 'manhattan')
#' summary
summary(BSTN_dist_man)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618
#' k-means clustering
BSTN_km <- stats::kmeans(BSTN_dist,
centers = 3)
#' visualize
pairs(BSTN_scale,
col = BSTN_km$cluster)
Now lets determine the optimal k value for k means clustering
# set.seed
set.seed(211)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <-kmeans(Boston,
centers = 2)
# plot the Boston dataset with clusters
pairs(BSTN_scale,
col = km$cluster)
From the q-plot we can see that the most drastic drop in “within cluster sum of squares (WCSS)” is at 2, which indicative of the optimal k value.
Super bonus
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)
## [1] 404 13
dim(BSTN_lda$scaling)
## [1] 13 3
# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% BSTN_lda$scaling
matrix_product <- as.data.frame(matrix_product)
plotly::plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)
The plots generally look similar with high crime rates clearly separated from the other groupings.
For this exercise we will be looking at human development index (HDI) parameters presented by the United nations developments programme to assess the development of countries. Details of the project and explanations of the variables can be found here.
First, lets load the data:
library(tidyverse)
#' read in modified file
human_data <- read.csv(here::here("data",
"human.csv"),
row.names = 1)
#' glimpse
dplyr::glimpse(human_data)
## Rows: 155
## Columns: 8
## $ Edu.sec.prop <dbl> 1.0072389, 0.9968288, 0.9834369, 0.9886128, 0.9690608, 0.~
## $ Lbr_prop <dbl> 0.8908297, 0.8189415, 0.8251001, 0.8840361, 0.8286119, 0.~
## $ Edu.Y <dbl> 17.5, 20.2, 15.8, 18.7, 17.9, 16.5, 18.6, 16.5, 15.9, 19.~
## $ Life.ExpAB <dbl> 81.6, 82.4, 83.0, 80.2, 81.6, 80.9, 80.9, 79.1, 82.0, 81.~
## $ GNI.pc <int> 64992, 42261, 56431, 44025, 45435, 43919, 39568, 52947, 4~
## $ MMR <int> 4, 6, 6, 5, 6, 7, 9, 28, 11, 8, 6, 4, 8, 4, 27, 2, 11, 6,~
## $ Adl_brthR <dbl> 7.8, 12.1, 1.9, 5.1, 6.2, 3.8, 8.2, 31.0, 14.5, 25.3, 6.0~
## $ Prl_prc <dbl> 39.6, 30.5, 28.5, 38.0, 36.9, 36.9, 19.9, 19.4, 28.2, 31.~
#' summary
base::summary(human_data)
## Edu.sec.prop Lbr_prop Edu.Y Life.ExpAB
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## GNI.pc MMR Adl_brthR Prl_prc
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.50
#' pairs
GGally::ggpairs(human_data)
From the summary we can observe the large differences between countries, especially with GNI. Form the visualization we can further see that education and life expectancy at birth are strongly positively correlated with GNI, while both maternal mortality rate and adolescent birth rate and negatively correlated.
Next, we carry out principle component analysis (PCA)
#' PCA
pca <- stats::prcomp(human_data)
biplot(pca,
choices = 1:2,
cex = c(0.8, 1),
col = c("grey40", "deeppink2"))
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped
Looks like most countries are clumped together but a a handful can be seen branching away towards the bottom and left. These might be extreme entries that deviate drastically from average.
Now, lets standardize the variables and try again
#' scale the data
human_data_s <- scale(human_data)
#' glimpse
base::summary(human_data_s)
## Edu.sec.prop Lbr_prop Edu.Y Life.ExpAB
## Min. :-2.8189 Min. :-2.6247 Min. :-2.7378 Min. :-2.7188
## 1st Qu.:-0.5233 1st Qu.:-0.5484 1st Qu.:-0.6782 1st Qu.:-0.6425
## Median : 0.3503 Median : 0.2316 Median : 0.1140 Median : 0.3056
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5958 3rd Qu.: 0.7350 3rd Qu.: 0.7126 3rd Qu.: 0.6717
## Max. : 2.6646 Max. : 1.6632 Max. : 2.4730 Max. : 1.4218
## GNI.pc MMR Adl_brthR Prl_prc
## Min. :-0.9193 Min. :-0.6992 Min. :-1.1325 Min. :-1.8203
## 1st Qu.:-0.7243 1st Qu.:-0.6496 1st Qu.:-0.8394 1st Qu.:-0.7409
## Median :-0.3013 Median :-0.4726 Median :-0.3298 Median :-0.1403
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.3712 3rd Qu.: 0.1932 3rd Qu.: 0.6030 3rd Qu.: 0.6127
## Max. : 5.6890 Max. : 4.4899 Max. : 3.8344 Max. : 3.1850
#' redo PCA analysis
pca2 <- stats::prcomp(human_data_s)
biplot(pca2,
choices = 1:2,
cex = c(0.8, 1),
col = c("grey40", "deeppink2"))
That makes things clearer. The countries are now more clearly distributed and the reason for separation is also much more obvious. The countries branching towards the left corner are mostly deviating due to education status while the ones branching towards the right corner are separated by maternal mortality rate and adolescent birth rate.
Moving things along. Tea! This data comes from the FactoMineR package and contains answers from a questionnaire filled by 300 individuals on their tea consumption, product preception, and personal questions. Details
library(FactoMineR)
#' load data
data("tea")
#' glimpse
dplyr::glimpse(tea)
## Rows: 300
## Columns: 36
## $ breakfast <fct> breakfast, breakfast, Not.breakfast, Not.breakfast, b~
## $ tea.time <fct> Not.tea time, Not.tea time, tea time, Not.tea time, N~
## $ evening <fct> Not.evening, Not.evening, evening, Not.evening, eveni~
## $ lunch <fct> Not.lunch, Not.lunch, Not.lunch, Not.lunch, Not.lunch~
## $ dinner <fct> Not.dinner, Not.dinner, dinner, dinner, Not.dinner, d~
## $ always <fct> Not.always, Not.always, Not.always, Not.always, alway~
## $ home <fct> home, home, home, home, home, home, home, home, home,~
## $ work <fct> Not.work, Not.work, work, Not.work, Not.work, Not.wor~
## $ tearoom <fct> Not.tearoom, Not.tearoom, Not.tearoom, Not.tearoom, N~
## $ friends <fct> Not.friends, Not.friends, friends, Not.friends, Not.f~
## $ resto <fct> Not.resto, Not.resto, resto, Not.resto, Not.resto, No~
## $ pub <fct> Not.pub, Not.pub, Not.pub, Not.pub, Not.pub, Not.pub,~
## $ Tea <fct> black, black, Earl Grey, Earl Grey, Earl Grey, Earl G~
## $ How <fct> alone, milk, alone, alone, alone, alone, alone, milk,~
## $ sugar <fct> sugar, No.sugar, No.sugar, sugar, No.sugar, No.sugar,~
## $ how <fct> tea bag, tea bag, tea bag, tea bag, tea bag, tea bag,~
## $ where <fct> chain store, chain store, chain store, chain store, c~
## $ price <fct> p_unknown, p_variable, p_variable, p_variable, p_vari~
## $ age <int> 39, 45, 47, 23, 48, 21, 37, 36, 40, 37, 32, 31, 56, 6~
## $ sex <fct> M, F, F, M, M, M, M, F, M, M, M, M, M, M, M, M, M, F,~
## $ SPC <fct> middle, middle, other worker, student, employee, stud~
## $ Sport <fct> sportsman, sportsman, sportsman, Not.sportsman, sport~
## $ age_Q <fct> 35-44, 45-59, 45-59, 15-24, 45-59, 15-24, 35-44, 35-4~
## $ frequency <fct> 1/day, 1/day, +2/day, 1/day, +2/day, 1/day, 3 to 6/we~
## $ escape.exoticism <fct> Not.escape-exoticism, escape-exoticism, Not.escape-ex~
## $ spirituality <fct> Not.spirituality, Not.spirituality, Not.spirituality,~
## $ healthy <fct> healthy, healthy, healthy, healthy, Not.healthy, heal~
## $ diuretic <fct> Not.diuretic, diuretic, diuretic, Not.diuretic, diure~
## $ friendliness <fct> Not.friendliness, Not.friendliness, friendliness, Not~
## $ iron.absorption <fct> Not.iron absorption, Not.iron absorption, Not.iron ab~
## $ feminine <fct> Not.feminine, Not.feminine, Not.feminine, Not.feminin~
## $ sophisticated <fct> Not.sophisticated, Not.sophisticated, Not.sophisticat~
## $ slimming <fct> No.slimming, No.slimming, No.slimming, No.slimming, N~
## $ exciting <fct> No.exciting, exciting, No.exciting, No.exciting, No.e~
## $ relaxing <fct> No.relaxing, No.relaxing, relaxing, relaxing, relaxin~
## $ effect.on.health <fct> No.effect on health, No.effect on health, No.effect o~
#' clean the data
tea_time <- tea %>%
dplyr::select(Tea,
How,
how,
sugar,
where,
lunch)
#' summaries
base::summary(tea_time)
## Tea How how sugar
## black : 74 alone:195 tea bag :170 No.sugar:155
## Earl Grey:193 lemon: 33 tea bag+unpackaged: 94 sugar :145
## green : 33 milk : 63 unpackaged : 36
## other: 9
## where lunch
## chain store :192 lunch : 44
## chain store+tea shop: 78 Not.lunch:256
## tea shop : 30
##
# visualize
ggplot(gather(tea_time), aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
## Warning: attributes are not identical across measure variables;
## they will be dropped
#' multple correspondence analysis (MCA)
mca <- MCA(tea_time, graph = FALSE)
#' summary of the model
summary(mca)
##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7
## Variance 0.279 0.261 0.219 0.189 0.177 0.156 0.144
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519 7.841
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953 77.794
## Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.141 0.117 0.087 0.062
## % of var. 7.705 6.392 4.724 3.385
## Cumulative % of var. 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr cos2
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139 0.003
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626 0.027
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111 0.107
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841 0.127
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979 0.035
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990 0.020
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347 0.102
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459 0.161
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968 0.478
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898 0.141
## v.test Dim.3 ctr cos2 v.test
## black 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 2.867 | 0.433 9.160 0.338 10.053 |
## green -5.669 | -0.108 0.098 0.001 -0.659 |
## alone -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 3.226 | 1.329 14.771 0.218 8.081 |
## milk 2.422 | 0.013 0.003 0.000 0.116 |
## other 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |
#' visualize MCA
plot(mca, invisible=c("ind"), habillage = "quali")
#' Biplot
factoextra::fviz_mca_biplot(mca, label = "var")
#' extra plots from the package
res.mca <- MCA(tea,
quanti.sup= 19,
quali.sup = 20:36,
graph = F)
plot(res.mca,
invisible = c("ind", "quali.sup", "quanti.sup"),
cex = 0.8)
## Warning: ggrepel: 30 unlabeled data points (too many overlaps). Consider
## increasing max.overlaps
From the plots it seems that Earl grey is usually taken with both milk and sugar, while black and green are not. Tea bags are likely to be bought from chain stores while unpackaged is bought from tea shops.
The last plot shows further categorical variables that provide further insights. Early grey seems to be the tea of choice for breakfast, lunch, pubs, and with friends, while green tea is usually reserved for dinner and privately.